Maurer-Cartan equation in the DGLA of graded derivations

Let M be a smooth manifold and D = ℒΨ+𝒥Ψ a solution of the Maurer-Cartan equation in the DGLA of graded derivations D* (M) of differential forms on M, where Ψ, Ψ are differential 1-form on M with values in the tangent bundle TM and ℒΨ, 𝒥Ψ are the d * and i * components of D. Under the hypothesis that IdT ( M ) + Ψ is invertible we prove that Ψ = b ( Ψ ) = - 1 2 _ ( I d T M + Ψ ) - 1 ∘ [ Ψ , Ψ ] ℱ 𝒩 {\rm{\Psi = }}b\left( {\rm{\Psi }} \right) = - {1 \over {}}{\left( {I{d_{TM}} + {\rm{\Psi }}} \right)^{ - 1}} \circ {\left[ {{\rm{\Psi }},{\rm{\Psi }}} \right]_{\mathcal{F}\mathcal{N}}} , where [·, ·]𝒡𝒩 is the Frölicher-Nijenhuis bracket. This yields to a classification of the canonical solutions e Ψ = ℒ Ψ +𝒥b ( Ψ ) of the Maurer-Cartan equation according to their type: e Ψ is of finite type r if there exists r∈ 𝒩 such that Ψr∘ [Ψ, Ψ]𝒡𝒩 = 0 and r is minimal with this property, where [·, ·]𝒡𝒩 is the Frölicher-Nijenhuis bracket. A distribution ξ ⊂TM of codimension k ⩾ 1 is integrable if and only if the canonical solution e Ψ associated to the endomorphism Ψ of TM which is trivial on ξ and equal to the identity on a complement of ξ in TM is of finite type ⩽ 1, respectively of finite type 0 if k = 1.

Effective construction of covers of canonical Hom-diagrams for equations over torsion-free hyperbolic groups

We show that, given a finitely generated group G as the coordinate group of a finite system of equations over a torsion-free hyperbolic group Γ, there is an algorithm which constructs a cover of a canonical solution diagram. The diagram encodes all homomorphisms from G to Γ as compositions of factorizations through Γ-NTQ groups and canonical automorphisms of the corresponding NTQ-subgroups. We also give another characterization of Γ-limit groups as iterated generalized doubles over Γ.

Global solvability and regularity for ∂¯ on an annulus between two weakly convex domains which satisfy property (P)

Let [Formula: see text] be a complex manifold of dimension [Formula: see text] and let [Formula: see text]. Let [Formula: see text] be a weakly [Formula: see text]-convex and [Formula: see text] be a weakly [Formula: see text]-convex in [Formula: see text] with smooth boundaries such that [Formula: see text]. Assume that [Formula: see text] and [Formula: see text] satisfy property [Formula: see text]. Then the compactness estimate for [Formula: see text]-forms [Formula: see text] holds for the [Formula: see text]-Neumann problem on the annulus domain [Formula: see text]. Furthermore, if [Formula: see text] is [Formula: see text]-closed [Formula: see text]-form, which is [Formula: see text] on [Formula: see text] and which is cohomologous to zero on [Formula: see text], the canonical solution [Formula: see text] of the equation [Formula: see text] is smooth on [Formula: see text].